Lecture A6
CEE 2219: STATICS AND INTRODUCTION TO MECHANICS OF MATERIALS
STRUCTURAL ANALYSIS
Dept. of Civil & Environ Engineering, SOE.
MR. DENIS MWABA MSc, B.Eng., R.Eng., PEIZ
CEE 2219 – STATICS & INTRODUCTION TO
MECHANICS OF MATERIALS
Lecture A6
 STRUCTURAL ANALYSIS
 PLANE TRUSSES (2D)
 SPACE TRUSSES (3D)
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LECTURE OBJECTIVES
 To show how to determine the
forces in the members of a truss
using the method of joints and
the method of sections.
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CHAPTER INTRODUCTION
• Structure refers a system of connected parts used to support loads, such as
buildings, bridges, dams, towers etc.
• Structural analysis is the prediction (design) of the performance of a given
structure under prescribed loads and/or other external effects such as support
movements and temperature changes
• Structural Engineers
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CHAPTER INTRODUCTION
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CHAPTER INTRODUCTION
• In the previous lecture we discussed the equilibrium of a rigid body i.e a
system of connected members treated as a single rigid body.
• We first drew a FBD of the body showing all forces external to the isolated
body, and then we applied the force and moment equations of equilibrium.
• In lecture A6 & A7 we will focus on the determination of the forces internal
to a structure - that is, forces of action and reaction between the connected
members.
• An engineering structure is any connected system of members built to
support or transfer forces and to safely withstand the loads applied to it.
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CHAPTER INTRODUCTION
• To determine the forces internal to an engineering structure, we must dismember
the structure and analyze separate FBD of individual members or combinations of
members
• This analysis requires careful application of Newton's third law, which states that
each action is accompanied by an equal and opposite reaction.
• In this chapter we will analyze the internal forces acting in several types of
structures (trusses, frames, and machines)
• In this treatment we consider only statically determinate structures, which do not
have more supporting constraints than are necessary to maintain an equilibrium
configuration
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CHAPTER INTRODUCTION
• The analysis of trusses, frames and machines, and
beams under concentrated loads constitutes a
straightforward application of the material developed
in the previous lectures. However, only trusses will be
discussed in this lecture
• The basic procedure developed in Chapter 3 and
Chapter 5 for isolating a body by constructing a
correct free-body diagram is essential for the analysis
of statically determinate structures.
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CHAPTER INTRODUCTION
• The analysis of trusses, frames and machines, and
beams under concentrated loads constitutes a
straightforward application of the material developed
in the previous lectures. However, only trusses will be
discussed in this lecture
• The basic procedure developed in Chapter 3 and
Chapter 5 for isolating a body by constructing a
correct free-body diagram is essential for the analysis
of statically determinate structures.
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PLANE TRUSSES
• A framework composed of members joined at their ends to form a rigid structure is
called a truss. Or a truss is a structure composed of slender members joined together
at their end points
• Bridges, roof supports, derricks, and other such structures common examples of
trusses.
• Structural members commonly used are; beams, channels, angles, bars, and special
shapes which are fastened together at their ends by welding, riveted connections, or
large bolts or pins.
• When the members of the truss lie essentially in a single plane, it is called a plane truss.
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PLANE TRUSSES
• The members commonly used in construction consist of wooden struts or metal bars.
• The truss shown in Fig. 6–1a is an example of a typical roof-supporting truss. In
this figure, the roof load is transmitted to the truss at the joints by means of a series of
purlins.
• Since this loading acts in the same plane as the truss, Fig. 6–1b, the analysis of the
forces developed in the truss members will be two-dimensional.
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PLANE TRUSSES
• In the case of a bridge, as shown, the load on the deck is
first transmitted to stringers, then to floor beams, and finally
to the joints of the two supporting side trusses. coplanar
• When bridge or roof trusses extend over large distances, a
rocker or roller is commonly used for supporting one end,
shown above
• This type of support allows freedom for expansion or
contraction of the members due to a change in temperature or
application of loads.
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PLANE TRUSSES
Assumptions for Design
• To design both the members and the connections of a truss, it is
necessary first to determine the force developed in each member when
the truss is subjected to a given loading.
• To do this we will make two important assumptions;
The members are assumed to be connected only by frictionless pins.
The loads must be applied at the joints
• Because of these two assumptions, each truss member will act as a
two-force member, and therefore the force acting at each end of the
member will be directed along the axis of the member.
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PLANE TRUSSES
Assumptions for Design
• If the force tends to elongate the member, it is a tensile
force (T), whereas if it tends to shorten the member, it is a
compressive force (C)
• In the actual design of a truss it is important to state whether
the nature of the force is tensile or compressive.
• Often, compression members must be made thicker than
tension members because of the buckling or column effect
that occurs when a member is in compression.
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PLANE TRUSSES
Simple Trusses
• If three members are pin connected at their ends, they form a
triangular truss that will be rigid, as shown below.
• Attaching two more members and connecting these members to a
new joint D forms a larger truss
• This procedure can be repeated as many times as desired to form an
even larger truss.
• If a truss can be constructed by expanding the basic triangular truss in
this way, it is called a simple truss
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PLANE TRUSSES
Simple Trusses
• Two methods for the force analysis of simple trusses will be
discussed.
• The free-body diagram of the truss as a whole is shown in
Fig. 4/6b
• The external reactions are usually determined first, by
applying the equilibrium equations to the truss as a whole
• Then the force analysis of the remainder of the truss is
performed
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SIMPLE TRUSSES
Method of Joints
• In order to analyze or design a truss, it is necessary to determine the force
in each of its members by using the method of joints.
• This method is based on the fact that if the entire truss is in equilibrium,
then each of its joints is also in equilibrium.
• Hence if the FBD of each joint is drawn, the force equilibrium equations
can then be used to obtain the member forces acting on each joint.
• Since the members of a plane truss are straight two-force members lying in
a single plane, each joint is subjected to a force system that is coplanar and
concurrent.
• As a result, only σ 𝐹𝑥 = 0 and σ 𝐹𝑦 = 0 need to be satisfied for equilibrium.
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SIMPLE TRUSSES
Method of Joints
• When using the method of joints, always start at a joint having at
least one known force and at most two unknown forces
• In this way, application of σ 𝐹𝑥 = 0 and σ 𝐹𝑦 = 0 yields two
algebraic equations which can be solved for the two unknowns.
• When applying these equations, the correct sense of an unknown
member force can be determined using one of the two possible
methods:
1) By inspection or 2) By assumption
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SIMPLE TRUSSES
Method of Joints
1) By inspection…..but in more complicated cases, the sense of an
unknown member force can be assumed; then, after applying the
equilibrium equations, the assumed sense can be verified from the
numerical results.
A positive answer indicates that the sense is correct, whereas a
negative answer indicates that the sense shown on the FBD must be
reversed.
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SIMPLE TRUSSES
Method of Joints
2) by assuming the unknown member forces acting on the joint’s FBD to
be in tension
If this is done, then numerical solution of the equilibrium equations will
yield positive scalars for members in tension and negative scalars for
members in compression.
Once an unknown member force is found, use its correct magnitude and
sense (T or C) on subsequent joint FBD
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SIMPLE TRUSSES
Example 6.1
Question
• Compute the force in each member of the loaded cantilever truss by the method of joints.
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SIMPLE TRUSSES
Example 6.1
Solution
• If it were not desired to calculate the external reactions at D and E, the analysis for a
cantilever truss could begin with the joint at the loaded end. However, this truss will be
analyzed completely, so the first step will be to compute the external forces at D and E
from the FBD of the truss as a whole. The equations of equilibrium give;
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SIMPLE TRUSSES
Example 6.1
Solution
• Next we draw FBD showing the forces acting on each of the
connecting pins.
• The correctness of the assigned directions of the forces is
verified when each joint is considered in sequence.
• There should be no question about the correct direction of the
forces on joint A
• Where T stands for tension and C stands for compression
JQ
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SIMPLE TRUSSES
Example 6.1
Solution
• Joint B must be analyzed next , since there are more than two
unknown forces on joint C.
• The force BC must provide an upward component, in which
case BD must balance the force to the left. Again the forces are
obtained from
• Joint C now contains only two unknowns, and these are
found in the same way as before:
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SIMPLE TRUSSES
Example 6.1
Solution
• Joint B must be analyzed next , since there
are more than two unknown forces on joint C.
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SIMPLE TRUSSES
Example 6.2
Question
• Determine the forces acting in all the members of the truss shown in Fig. below.
JQ
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SIMPLE TRUSSES
Example 6.2
Solution
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SIMPLE TRUSSES
Example 6.2
Solution
JQ
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2022-
SIMPLE TRUSSES
Example 6.3
Question
• Determine the force in each member of the truss. State if the members
are in tension or compression.
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SIMPLE TRUSSES
Example 6.3
Solution
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SIMPLE TRUSSES
Zero-Force Members
• Truss analysis using the method of joints is greatly simplified if we
can first identify those members which support no loading.
• These zero-force members are used to increase the stability of the truss
during construction and to provide added support if the loading is
changed.
• The zero-force members of a truss can generally be found by
inspection of each of the joints.
• For example, consider the truss shown in Fig. 6–11a. If a FBD of the
pin at joint A is drawn, Fig. 6–11b, it is seen that members AB and AF
are zero-force members.
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SIMPLE TRUSSES
Zero-Force Members
• Note that we could not have come to this conclusion if we had
considered the FBDs of joints F or B simply because there are
five unknowns at each of these joints.)
• In a similar manner, consider the FBD of joint D, Fig. 6–11c.
Here again it is seen that DC and DE are zero-force members.
• From these observations, we can conclude that if only two
non-collinear members form a truss joint and no external load or
support reaction is applied to the joint, the two members must
be zero-force members.
• The load on the truss in Fig. 6–11a is therefore supported by
only five members as shown in Fig. 6–11d.
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SIMPLE TRUSSES
Zero-Force Members
• Now consider the truss shown in Fig. a. The FBD of the pin at joint D is
shown in Fig. b.
• By orienting the y axis along members DC and DE and the x axis along
member DA, it is seen that DA is a zero-force member.
• Note that this is also the case for member CA, Fig. c.
• In general then, if three members form a truss joint for which two of the
members are collinear, the third member is a zero-force member provided
no external force or support reaction has a component that acts along this
member.
• The truss shown in Fig. 6–12d is therefore suitable for supporting the load
P.
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STRUCTURAL ANALYSIS - TRUSSES
BREAK FOR REVIEW
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THE END
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